1. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay exponential function baseasymptote exponential growth and decay Vocabulary Write and evaluate exponential expressions to model growth and decay situations. Objective

2. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Notes In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year. 1. Write a function for world population. Does the function represent growth or decay? 2. Predict the population in 2020. The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? 4. Predict the value in 4 years.

3. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Growth that doubles every year can be modeled by using a function with a variable as an exponent. This function is known as an exponential function. The parent exponential function is f(x) = b x, where the base b is a constant and the exponent x is the independent variable.

4. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay The graph of the parent function f(x) = 2 x is shown. The domain is all real numbers and the range is {y|y > 0}.

5. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Notice as the x-values decrease, the graph of the function gets closer and closer to the x-axis. The function never reaches the x-axis because the value of 2 x cannot be zero. In this case, the x-axis is an asymptote. An asymptote is a line that a graphed function approaches as the value of x gets very large or very small.

6. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay A function of the form f(x) = ab x, with a > 0 and b > 1, is an exponential growth function, which increases as x increases. When 0 < b < 1, the function is called an exponential decay function, which decreases as x increases.

7. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Negative exponents indicate a reciprocal. For example: Remember! In the function y = b x, y is a function of x because the value of y depends on the value of x. Remember!

8. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Then graph. Example 1A: Graphing Exponential Functions Step 1 Find the value of the base. The base,,is less than 1. This is an exponential decay function.

9. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Example 1A Continued Step 2 Graph the function by using a table of values. x024681012 f(x)105.63.21.81.00.60.3

10. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. Then graph. Example 1B: Graphing Exponential Functions Step 1 Find the value of the base. g(x) = 100(1.05) x The base, 1.05, is greater than 1. This is an exponential growth function. g(x) = 100(1.05) x

11. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Tell whether the function p(x) = 5(1.2 x ) shows growth or decay. Then graph. Step 1 Find the value of the base. Example 1C p(x) = 5(1.2 x ) The base, 1.2, is greater than 1. This is an exponential growth function.

12. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Step 2 Graph the function by using a table of values. x–12–8–404810 f(x)0.561.22.4510.421.530.9 Example 1C Continued

13. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay You can model growth or decay by a constant percent increase or decrease with the following formula: In the formula, the base of the exponential expression, 1 + r, is called the growth factor. Similarly, 1 – r is the decay factor.

14. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years? Example 2A: Economics Application Step 1 Write a function to model the growth in value of her investment. f(t) = a(1 + r) t Exponential growth function. Substitute 5000 for a and 0.0625 for r. f(t) = 5000(1 + 0.0625) t f(t) = 5000(1.0625) t Simplify.

15. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Clara invests $5000 in an account that pays 6.25% interest per year. What will her investment be worth in seven years? Example 2A: Economics Application Step 2 Use function to evaluate at t = 7. f(t) = 5000(1.0625) t Substitute 7 for t.

16. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay In 1981, the Australian humpback whale population was 350 and increased at a rate of 14% each year since then. Write a function to model population growth. P(t) = a(1 + r) t Substitute 350 for a and 0.14 for r. P(t) = 350(1 + 0.14) t P(t) = 350(1.14) t Simplify. Exponential growth function. Example 2B

17. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay A city population, which was initially 15,500, has been dropping 3% a year. Write an exponential function. What will the population be after ten years? Example 3: Depreciation Application f(t) = a(1 – r) t Substitute 15,500 for a and 0.03 for r. f(t) = 15,500(1 – 0.03) t f(t) = 15,500(0.97) t Simplify. Exponential decay function. f(t) = 15,500(0.97) 10 =11,430 t=10 for ten years.

18. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay Notes In 2000, the world population was 6.08 billion and was increasing at a rate 1.21% each year. 1. Write a function for world population. Does the function represent growth or decay? P(t) = 6.08(1.0121) t 2. Use a graph to predict the population in 2020. ≈ 7.73 billion The value of a $3000 computer decreases about 30% each year. 3. Write a function for the computer’s value. Does the function represent growth or decay? 4. Use a graph to predict the value in 4 years. V(t)≈ 3000(0.7) t ≈ $720.30

19. Holt Algebra 2 7-1 Exponential Functions, Growth, and Decay 5. Tell whether the function shows growth or decay. Then graph. Notes (continued) f(x) = 8(½) x